Recognition of Mathematics Notation Via Computer Using Baseline

Total Page:16

File Type:pdf, Size:1020Kb

Recognition of Mathematics Notation Via Computer Using Baseline Recognition of Mathematics Notation via Computer Using Baseline Structure Richard Zanibbi zanibbicsqueensuca August External Technical Rep ort ISSN Department of Computing and Information Science Queens University Kingston Ontario Canada KL N Do cument prepared August c Copyright Richard Zanibbi Abstract The spatial structure of mathematical expressions may b e represented using the structure of baselines in an expression This baseline structure may b e represented as a tree and then rewritten to represent dierent interpretations of spatial structure in an expression This structure tree may also b e trans A lated into mathematical string languages such as L T X Maple etc A parsing E algorithm which extracts baseline structure in mathematics expressions from a list of attributed symb ols is presented along with an implementation of the algorithm The implementation of the parsing algorithm has b een in terfaced with an online mathematics entry system The resulting integrated system allowed testing of the parser on hundreds of input expressions with encouraging results Future work in the extension of the current mathematics recognition technique and application of directionbased recognition to other notations such as music notation and circuit diagrams are also discussed Acknowledgments Thanks to Dorothea Blostein who was a fun knowledgeable and supp ortive sup ervisor Ed Lank for all his help with the design of DRACULAE and the ideas in this thesis Nick Willan for his implementation of the user in terface for DRACULAE and patiently listening to me work out many of the ideas in this thesis at the white b oard for the record Nick named the parsing application DRACULA and I simply added an E Ken Whelan for his fantastic pro ofreading Gary Anderson who saved my life on many an o ccasion and was consistently an amazing source of information useful and entertaining Ho da Fahmy whose exp erience and insight have b een very helpful and who raised some of the semantics issues discussed at the end of this thesis Jianping Wu for his help with course work and things C Talib Hussein Laurie Ricker and all the other students and professors who help ed me survive my MSc in various ways Thanks to Genarro Costagliola who provided what for me was a highly inuential draft pap er on recognizing a subset of mathematics notation using a p ositional grammar and some valuable comments Thanks to Steve Smithies Kevin Novins and Jim Arvo for letting me use FFES to develop DRACULAE Without FFES it would have b een much harder to develop and test DRACULAE Thanks to Debby Linda Irene Sandra Tom Bradshaw and Gary Powley for all the help with the hard thankless stu that keeps things going and gets them done Finally thanks to Katey for listening to my rambles ab out this stu over and over again and for b eing a fabulous pro ofreader wife and friend Chapter Intro duction In certain domains the ability to use visual languages to communicate with computers allows simpler and more succinct expression of a users intentions than is p ossible with a string language For example it is easier for a user to draw a complicated mathematical expression in mathematics notation a visual language than to write the same expression in a string language A such as L T X or Maple Due to the sometimes more succinct and intuitive E expressions available in visual languages in comparison to string languages disciplines such as music architecture and engineering commonly use visual languages to convey information in the form of diagrams Informally a visual language is comprised of sets of symb ols laid out in two or three dimensions diagrams which comprise the set of legal expressions under a syntactic and semantic denition of the visual language The principal dierence b etween a visual language and a string language is that the syntax of a visual language is of higher dimensionality As a result b o okkeeping is required for spatial relationships b etween the terminals and nonterminals of the language when parsing A visual language may b e dened as in the following Graphical Primitive A graphical primitive is a line or dot in an image Symb ol A symb ol is dened by a nonempty set of spatial rela tions on graphical primitives which are intended to b e p er ceived as a single unit Diagram A diagram is a two or three dimensional collection set of symb ols Intro duction Visual Language A visual language is a set of diagrams which are considered valid expressions in the dened language Both the syntax and semantics of a visual language are de termined through the spatial relationships b etween symb ols in a diagram based on description in Currently there is ongoing research into the design and development of visual programming languages including compilercompilers for these lan guages The diagrams of a visual programming language may b e translated to an executable form using a visual language compiler Spread sheets are the most common and p erhaps also the most successful typ e of visual programming language to date Some visual languages such as math notation music notation and engi neering drawings are not formally dened and have dialects ie notational practices which are not standard but accepted We call these visual lan guages natural visual languages and dene them as in the following Dialect A dialect of a visual language is a variant of a visual language which employs symb olic andor spatial conventions which are not standard in all other variants of the visual language Natural Visual Language A natural visual language is a vi sual language for which no formal description exists andor dialects of the visual language exist Mathematics notation is the natural visual language on which we fo cus in this thesis For the remainder of this thesis when mathematics nota tion is used it is the Western standard read lefttoright which is b eing indicated for an interesting comparison see for a discussion of recogniz ing Arabic mathematical notation which is read righttoleft Mathematics notation is not formally dened we usually assume that mathematics no tation includes notations for algebra calculus logic and a numb er of other mathematical disciplines Mathematical discourse also often involves the use of dened notation and symb ols often have dierent semantic interpreta tions eg fb may b e interpreted as function application or as implied multiplication or may b e dened notation pro ducing dialects Natural visual languages such as mathematics notation have b oth hard and soft conventions Hard conventions are those asp ects of a visual lan guage that are consistent across dialects Soft conventions may or may not Intro duction apply to a particular diagram dep ending on the dialect used when a diagram was created An example of a hard convention in mathematics notation is the lefttoright direction of interpretation Soft conventions in mathematics notation include layout of symb ols The lack of formal denition for natural visual languages necessitates identication of hard and soft constraints b efore a language denition may b e constructed Without formal denitions it is dicult and p erhaps im p ossible to pro duce a completely accurate list of the two sets of conventions for any natural visual language as these have to b e obtained largely through observation and introsp ection Once the conventions of the language are felt to b e reasonably well understo o d and classied as hard or soft one may set ab out dening a syntax sp ecifying legal symb ol placement in diagrams and the semantic interpretation of this set of legal diagrams Again the lack of formal denition prevents this language denition from b eing complete in many cases In the case of mathematics notation some general descriptions of the structure of the notation are available from typ esetting pro cedures a history of the evolution of mathematics notation and a history of mathematics Ca joris history of notation unfortunately fo cuses on the evolution of symb ols rather than on the use of space b etween symb ols In addition the typ esetting references describ e notation from the p ersp ective of generating mathematics notation and do not explicitly state the hard and soft conventions of the notation Dialects complicate syntactic and semantic denition due to the resulting multiple interpretations that are p ossible For example in mathematics no tation it is imp ossible to determine whether cos should b e interpreted as a function or implied multiplication of variables without a prior decision ab out how to interpret this letter grouping ie cho osing a soft convention This is an example of how interpreting visual language dialects requires a priori information ab out the intended dialect of interpretation b efore the intended syntax and semantics can b e obtained from a diagram If this information is available the appropriate soft conventions may b e adopted to prop erly recognize a diagram Intro duction The Visual Language Recognition Pro cess Figure shows the pro cess of recognizing visual languages by computer and the four general typ es of diagram representation usable by a computer lab eled Typ es IIV Each of these diagrammatic representations is explained further b elow Visual Language Editor - Allows direct input of symbols TYPE II: Attributed Symbol List via mouse, data tablet and keyboard. TYPE IV: Semantic Representation Parsing TYPE I: Pixel Map Symbol Recognition a. Handwritten, Noisy Semantic Evaluation b. Handwritten, Noise-Free TYPE III:Structural Representation c. Typeset, Noisy * d. Typeset, Noise-Free
Recommended publications
  • Writing Mathematical Expressions in Plain Text – Examples and Cautions Copyright © 2009 Sally J
    Writing Mathematical Expressions in Plain Text – Examples and Cautions Copyright © 2009 Sally J. Keely. All Rights Reserved. Mathematical expressions can be typed online in a number of ways including plain text, ASCII codes, HTML tags, or using an equation editor (see Writing Mathematical Notation Online for overview). If the application in which you are working does not have an equation editor built in, then a common option is to write expressions horizontally in plain text. In doing so you have to format the expressions very carefully using appropriately placed parentheses and accurate notation. This document provides examples and important cautions for writing mathematical expressions in plain text. Section 1. How to Write Exponents Just as on a graphing calculator, when writing in plain text the caret key ^ (above the 6 on a qwerty keyboard) means that an exponent follows. For example x2 would be written as x^2. Example 1a. 4xy23 would be written as 4 x^2 y^3 or with the multiplication mark as 4*x^2*y^3. Example 1b. With more than one item in the exponent you must enclose the entire exponent in parentheses to indicate exactly what is in the power. x2n must be written as x^(2n) and NOT as x^2n. Writing x^2n means xn2 . Example 1c. When using the quotient rule of exponents you often have to perform subtraction within an exponent. In such cases you must enclose the entire exponent in parentheses to indicate exactly what is in the power. x5 The middle step of ==xx52− 3 must be written as x^(5-2) and NOT as x^5-2 which means x5 − 2 .
    [Show full text]
  • The Recent Trademarking of Pi: a Troubling Precedent
    The recent trademarking of Pi: a troubling precedent Jonathan M. Borwein∗ David H. Baileyy July 15, 2014 1 Background Intellectual property law is complex and varies from jurisdiction to jurisdiction, but, roughly speaking, creative works can be copyrighted, while inventions and processes can be patented, and brand names thence protected. In each case the intention is to protect the value of the owner's work or possession. For the most part mathematics is excluded by the Berne convention [1] of the World Intellectual Property Organization WIPO [12]. An unusual exception was the successful patenting of Gray codes in 1953 [3]. More usual was the carefully timed Pi Day 2012 dismissal [6] by a US judge of a copyright infringement suit regarding π, since \Pi is a non-copyrightable fact." We mathematicians have largely ignored patents and, to the degree we care at all, have been more concerned about copyright as described in the work of the International Mathematical Union's Committee on Electronic Information and Communication (CEIC) [2].1 But, as the following story indicates, it may now be time for mathematicians to start paying attention to patenting. 2 Pi period (π:) In January 2014, the U.S. Patent and Trademark Office granted Brooklyn artist Paul Ingrisano a trademark on his design \consisting of the Greek letter Pi, followed by a period." It should be noted here that there is nothing stylistic or in any way particular in Ingrisano's trademark | it is simply a standard Greek π letter, followed by a period. That's it | π period. No one doubts the enormous value of Apple's partly-eaten apple or the MacDonald's arch.
    [Show full text]
  • Fundamentals of Mathematics Lecture 3: Mathematical Notation Guan-Shieng Fundamentals of Mathematics Huang Lecture 3: Mathematical Notation References
    Fundamentals of Mathematics Lecture 3: Mathematical Notation Guan-Shieng Fundamentals of Mathematics Huang Lecture 3: Mathematical Notation References Guan-Shieng Huang National Chi Nan University, Taiwan Spring, 2008 1 / 22 Greek Letters I Fundamentals of Mathematics 1 A, α, Alpha Lecture 3: Mathematical 2 B, β, Beta Notation Guan-Shieng 3 Γ, γ, Gamma Huang 4 ∆, δ, Delta References 5 E, , Epsilon 6 Z, ζ, Zeta 7 H, η, Eta 8 Θ, θ, Theta 9 I , ι, Iota 10 K, κ, Kappa 11 Λ, λ, Lambda 12 M, µ, Mu 2 / 22 Greek Letters II Fundamentals 13 of N, ν, Nu Mathematics Lecture 3: 14 Ξ, ξ, Xi Mathematical Notation 15 O, o, Omicron Guan-Shieng Huang 16 Π, π, Pi References 17 P, ρ, Rho 18 Σ, σ, Sigma 19 T , τ, Tau 20 Υ, υ, Upsilon 21 Φ, φ, Phi 22 X , χ, Chi 23 Ψ, ψ, Psi 24 Ω, ω, Omega 3 / 22 Logic I Fundamentals of • Conjunction: p ∧ q, p · q, p&q (p and q) Mathematics Lecture 3: Mathematical • Disjunction: p ∨ q, p + q, p|q (p or q) Notation • Conditional: p → q, p ⇒ q, p ⊃ q (p implies q) Guan-Shieng Huang • Biconditional: p ↔ q, p ⇔ q (p if and only if q) References • Exclusive-or: p ⊕ q, p + q • Universal quantifier: ∀ (for all) • Existential quantifier: ∃ (there is, there exists) • Unique existential quantifier: ∃! 4 / 22 Logic II Fundamentals of Mathematics Lecture 3: Mathematical Notation Guan-Shieng • p → q ≡ ¬p ∨ q Huang • ¬(p ∨ q) ≡ ¬p ∧ ¬q, ¬(p ∧ q) ≡ ¬p ∨ ¬q References • ¬∀x P(x) ≡ ∃x ¬P(x), ¬∃x P(x) ≡ ∀x ¬P(x) • ∀x ∃y P(x, y) 6≡ ∃y ∀x P(x, y) in general • p ⊕ q ≡ (p ∧ ¬q) ∨ (¬p ∧ q) 5 / 22 Set Theory I Fundamentals of Mathematics • Empty set: ∅, {} Lecture 3: Mathematical • roster: S = {a , a ,..., a } Notation 1 2 n Guan-Shieng defining predicate: S = {x| P(x)} where P is a predicate Huang recursive description References • N: natural numbers Z: integers R: real numbers C: complex numbers • Two sets A and B are equal if x ∈ A ⇔ x ∈ B.
    [Show full text]
  • Table of Mathematical Symbols = ≠ <> != < > ≪ ≫ ≤ <=
    Table of mathematical symbols - Wikipedia, the free encyclopedia Página 1 de 13 Table of mathematical symbols From Wikipedia, the free encyclopedia For the HTML codes of mathematical symbols see mathematical HTML. Note: This article contains special characters. The following table lists many specialized symbols commonly used in mathematics. Basic mathematical symbols Name Symbol Read as Explanation Examples Category equality x = y means x and y is equal to; represent the same 1 + 1 = 2 = equals thing or value. everywhere ≠ inequation x ≠ y means that x and y do not represent the same thing or value. is not equal to; 1 ≠ 2 does not equal (The symbols != and <> <> are primarily from computer science. They are avoided in != everywhere mathematical texts. ) < strict inequality x < y means x is less than y. > is less than, is x > y means x is greater 3 < 4 greater than, is than y. 5 > 4. much less than, is much greater x ≪ y means x is much 0.003 ≪ 1000000 ≪ than less than y. x ≫ y means x is much greater than y. ≫ order theory inequality x ≤ y means x is less than or equal to y. ≤ x ≥ y means x is <= is less than or greater than or equal to http://en.wikipedia.org/wiki/Table_of_mathematical_symbols 16/05/2007 Table of mathematical symbols - Wikipedia, the free encyclopedia Página 2 de 13 equal to, is y. greater than or The symbols and equal to ( <= 3 ≤ 4 and 5 ≤ 5 ≥ >= are primarily from 5 ≥ 4 and 5 ≥ 5 computer science. They >= order theory are avoided in mathematical texts.
    [Show full text]
  • Outline of Mathematics
    Outline of mathematics The following outline is provided as an overview of and 1.2.2 Reference databases topical guide to mathematics: • Mathematical Reviews – journal and online database Mathematics is a field of study that investigates topics published by the American Mathematical Society such as number, space, structure, and change. For more (AMS) that contains brief synopses (and occasion- on the relationship between mathematics and science, re- ally evaluations) of many articles in mathematics, fer to the article on science. statistics and theoretical computer science. • Zentralblatt MATH – service providing reviews and 1 Nature of mathematics abstracts for articles in pure and applied mathemat- ics, published by Springer Science+Business Media. • Definitions of mathematics – Mathematics has no It is a major international reviewing service which generally accepted definition. Different schools of covers the entire field of mathematics. It uses the thought, particularly in philosophy, have put forth Mathematics Subject Classification codes for orga- radically different definitions, all of which are con- nizing their reviews by topic. troversial. • Philosophy of mathematics – its aim is to provide an account of the nature and methodology of math- 2 Subjects ematics and to understand the place of mathematics in people’s lives. 2.1 Quantity Quantity – 1.1 Mathematics is • an academic discipline – branch of knowledge that • Arithmetic – is taught at all levels of education and researched • Natural numbers – typically at the college or university level. Disci- plines are defined (in part), and recognized by the • Integers – academic journals in which research is published, and the learned societies and academic departments • Rational numbers – or faculties to which their practitioners belong.
    [Show full text]
  • List of Mathematical Symbols by Subject from Wikipedia, the Free Encyclopedia
    List of mathematical symbols by subject From Wikipedia, the free encyclopedia This list of mathematical symbols by subject shows a selection of the most common symbols that are used in modern mathematical notation within formulas, grouped by mathematical topic. As it is virtually impossible to list all the symbols ever used in mathematics, only those symbols which occur often in mathematics or mathematics education are included. Many of the characters are standardized, for example in DIN 1302 General mathematical symbols or DIN EN ISO 80000-2 Quantities and units – Part 2: Mathematical signs for science and technology. The following list is largely limited to non-alphanumeric characters. It is divided by areas of mathematics and grouped within sub-regions. Some symbols have a different meaning depending on the context and appear accordingly several times in the list. Further information on the symbols and their meaning can be found in the respective linked articles. Contents 1 Guide 2 Set theory 2.1 Definition symbols 2.2 Set construction 2.3 Set operations 2.4 Set relations 2.5 Number sets 2.6 Cardinality 3 Arithmetic 3.1 Arithmetic operators 3.2 Equality signs 3.3 Comparison 3.4 Divisibility 3.5 Intervals 3.6 Elementary functions 3.7 Complex numbers 3.8 Mathematical constants 4 Calculus 4.1 Sequences and series 4.2 Functions 4.3 Limits 4.4 Asymptotic behaviour 4.5 Differential calculus 4.6 Integral calculus 4.7 Vector calculus 4.8 Topology 4.9 Functional analysis 5 Linear algebra and geometry 5.1 Elementary geometry 5.2 Vectors and matrices 5.3 Vector calculus 5.4 Matrix calculus 5.5 Vector spaces 6 Algebra 6.1 Relations 6.2 Group theory 6.3 Field theory 6.4 Ring theory 7 Combinatorics 8 Stochastics 8.1 Probability theory 8.2 Statistics 9 Logic 9.1 Operators 9.2 Quantifiers 9.3 Deduction symbols 10 See also 11 References 12 External links Guide The following information is provided for each mathematical symbol: Symbol: The symbol as it is represented by LaTeX.
    [Show full text]
  • Towards a Better Notation for Mathematics
    Towards a Better Notation for Mathematics Christopher Olah ([email protected]) July 8, 2010 Abstract: This paper explores alternative notations for mathe- matics. We consider numerical notations that make arithmetic eas- ier, a notation for quantifiers that reduces the number of variables one needs to track, and many other ideas. (Regarding Notation: The typographical quality of new sym- bols introduced in this paper is terribly lacking { you will likely need to zoom in on them to see them well. This is because the author is injecting them as images. A font is in the works but isn't much better, as the author is not a skilled font-craft.) Update: I've made some minor changes to young-me's work to clean things up a bit. That said, please keep in mind that I've progressed a long way since writing this and it isn't representative of my current abilities. Contents 1 Introduction2 2 Version Representation3 3 Numbers & Vectors3 4 Sets5 5 Boolean Operations6 6 Standard Functions6 7 Trigonometry7 8 Calculus8 9 Quantifiers8 1 10 Science9 11 Equality, Scope & Algebra9 1 Introduction When one considers how complicated the ideas mathematical notation must represent are, it clearly does quite a good job. Despite this, the author believes that any claim that modern mathematical notation is the best should be met with extreme scepticism. Mathematical notation is a natural language: no one sat down and con- structed it, but rather it formed gradually by people making changes that are adopted. Most changes are not adopted; whether they are depends on a variety of factors including: mathematical utility, ease of adoption (the average individ- ual doesn't want to spend hours learning), dissemination, and shear dumb luck.
    [Show full text]
  • Unicode Nearly Plain Text Encoding of Mathematics
    Unicode Nearly Plain Text Encoding of Mathematics Unicode Nearly Plain-Text Encoding of Mathematics Murray Sargent III Office Authoring Services, Microsoft Corporation 28-Aug-06 1. Introduction ............................................................................................................ 2 2. Encoding Simple Math Expressions ...................................................................... 3 2.1 Fractions .......................................................................................................... 4 2.2 Subscripts and Superscripts........................................................................... 6 2.3 Use of the Blank (Space) Character ............................................................... 7 3. Encoding Other Math Expressions ........................................................................ 8 3.1 Delimiters ........................................................................................................ 8 3.2 Literal Operators ........................................................................................... 10 3.3 Prescripts and Above/Below Scripts .......................................................... 11 3.4 n-ary Operators ............................................................................................. 11 3.5 Mathematical Functions ............................................................................... 12 3.6 Square Roots and Radicals ........................................................................... 13 3.7 Enclosures ....................................................................................................
    [Show full text]
  • Distinguishing Mathematics Notation from English Text Using Computational Geometry
    Distinguishing Mathematics Notation from English Text using Computational Geometry Derek M. Drake Henry S. Baird Lehigh University, CSE Dept. Lehigh University, CSE Dept. Bethlehem, PA 18015 USA Bethlehem, PA 18015 USA [email protected] [email protected] Abstract and browsers of the OCR results as collections of images uncorrupted by OCR errors, for example in “visual math A trainable method for distinguishing between mathe- indexes” to assist fast search. Also, since several experi- matics notation and natural language (here, English) in mental systems exist to recognize mathematics [10, 5, 3] images of textlines, using computational geometry meth- which run independently of commercial OCR machines, au- ods only with no assistance from symbol recognition, is de- tomatic isolation of math images could allow them to be ap- scribed. The input to our method is a “neighbor graph” plied selectively where they are most likely to succeed. The extracted from a bilevel image of an isolated textline by the Infty system [12] is an example of such an integration. method of Kise [8]: this is a pruned form of Delaunay tri- Mathematics notation is often largely—though seldom angulation of the set of locations of black connected com- entirely—independent of the natural language of the sur- ponents. Our method first attempts to classify each vertex rounding text, and so there may be important uses for a and, separately, each edge of the neighbor graph as belong- “language-free” mathematics recognition system applicable ing to math or English; then these results are combined to independently of any language-specific OCR system.
    [Show full text]
  • A Guide to Writing Mathematics
    A Guide to Writing Mathematics Dr. Kevin P. Lee Introduction This is a math class! Why are we writing? There is a good chance that you have never written a paper in a math class before. So you might be wondering why writing is required in your math class now. The Greek word mathemas, from which we derive the word mathematics, embodies the notions of knowledge, cognition, understanding, and perception. In the end, mathematics is about ideas. In math classes at the university level, the ideas and concepts encountered are more complex and sophisticated. The mathematics learned in college will include concepts which cannot be expressed using just equations and formulas. Putting mathemas on paper will require writing sentences and paragraphs in addition to the equations and formulas. Mathematicians actually spend a great deal of time writing. If a mathematician wants to contribute to the greater body of mathematical knowledge, she must be able communicate her ideas in a way which is comprehensible to others. Thus, being able to write clearly is as important a mathematical skill as being able to solve equations. Mastering the ability to write clear mathematical explanations is important for non-mathematicians as well. As you continue taking math courses in college, you will come to know more mathematics than most other people. When you use your mathematical knowledge in the future, you may be required to explain your thinking process to another person (like your boss, a co-worker, or an elected official), and it will be quite likely that this other person will know less math than you do.
    [Show full text]
  • Mathematical Symbols and Abbreviations Mccp-Matthews-Symbols-001 This Leaflet Provides Information on Symbols and Notation Commonly Used in Mathematics
    community project Mathematical Symbols and Abbreviations mccp-matthews-symbols-001 This leaflet provides information on symbols and notation commonly used in mathematics. It is designed to enable further information to be found from resources in mathcentre (www.mathcentre.ac.uk). In the table below, the symbol or notation is given in column one. It is not always obvious how the combination of characters used in mathematical notation is said, so where appropriate this information is given in column two. Column three explains the use of the symbol and an example may be given for further explanation in column four. The last column contains a phrase to be entered as a search topic in mathcentremathcentre if further details are required. community project Care should be taken as context is important.encouraging Identical math academicsematical to symbols share and maths notation support are used resources in different circumstances to convey very different ideas. All mccp resources are released under a Creative Commons licence Symbol Say Means Example mathcentre Search Topic P sigma Represents summation “Sigma notation” Parallel In the same direction as “The gradient of a straight k line segment” || Modulus, absolute The size of a number, ignoring the sign |3|= 3, | 3|= 3 Modulus (vertical lines, ei- value − ther side of a num- ber or variable) |A| Determinant of Determinant of matrix A Determinants matrix A det (A) Determinant of Determinant of matrix A Determinants matrix A c Janette Matthews Tony Croft www.mathcentre.ac.uk Loughborough University Loughborough University community project Mathematical Symbols and Abbreviations (continued) Symbol Say Means Example mathcentre Search Topic () brackets Used in many different contexts e.g.
    [Show full text]
  • Mathematics Review for Science Students
    Mathematics Review for Science Students Math Department Math Study Center Grossmont College 1. 0 - Exponents and Scientific Notation 1.1 - Exponential Notation Exponential notation is a simple mathematical notation to represent repeated multiplication of the same number. Numbers in exponential notation are written in the form x n , where x is the base and n is the exponent. Examples: 1. 34 = 3×3×3×3 = 81 2. 103 =10×10×10 =1000 1.2 - Some Rules of Exponents m n m+n 1. x x x −n 1 ⋅ = 3. x = for x ≠ 0 x n m x m−n 2. = x 0 x n 4. x =1 for x ≠ 0 Examples: 3 4 3+4 7 1. 10 10 10 10 −2 1 1 × = = 5. 3 = = 32 9 2. 10−2 ×105 =10−2+5 =103 1 1 1 6. 10−4 = = = 106 104 10⋅10⋅10⋅10 10000 3. =106−3 =103 103 7. 40 =1 7 10 7−(−4) 7+4 11 4. =10 =10 =10 0 10−4 8. 10 =1 1.3 - Scientific Notation Scientific notation is shorthand for writing very large and very small numbers. Proper scientific notation is written using powers of 10 and is in the form N ×10m where N is a number greater than or equal to 1 but less than 10 (1 ≤ N <10 ) and m is the exponent. For example, the speed of light,1.86 ×105 miles per second is in scientific notation but 18.6 ×104 is not since N which is 18.6 is not between 1 and 10.
    [Show full text]